ART. III. 1. Théorie de la double Réfraction de la Lumière. Par E. L. Malus. 4to. Par. 1810. Pp. 302; with 3 Plates. 2. Mémoire sur de nouveaux Rapports entre la Réflexion et la Polarisation de la Lumière. Par M. Biot. Lu à l'Institut le 1 Juin, 1812. Pp. 152; with 1 Plate.

3. Versuche über Spiegelung und Brechung. Experiments on the Reflection and Refraction of Light. By Dr. Seebeck. Schweigger's Journ.. Nuremberg, 1813; with a coloured Plate. 4. A Treatise on new Philosophical Instruments, with Experiments on Light and Colours. By David Brewster, LL. D. 8vo. Edinb. 1813. Pp. 442; with 12 Plates.

Technineteenders xt unnecessary to make any apology for inHE intimate connexion of the subjects of these works with cluding our account of them in one article; since the greater number of the observations which they contain have arisen more or less immediately from the prosecution, in the different parts of Europe, of the important discoveries of Mr. Malus, respecting the properties exhibited by light which has been subjected to oblique reflection or refraction. Of these discoveries we have already given some account in our sixth number (p. 472); and the honourable testimonials of public approbation, which their author has since received, in particular from the Royal Society of London, as well as from the Institute of France, sufficiently show that our estimate of his merits was not exaggerated. Most unfortunately for the sciences, his career has been cut short by an early death, in the midst of his researches and improvements; but this event did not take place, as Dr. Brewster seems to imagine, so immediately after the adjudication of Count Rumford's medal, as to have rendered him incapable of being informed of the honour that was conferred on him, and of appreciating its value.

In the present work of Mr. Malus, there is less of absolute novelty than of minute and interesting research, upon a point, respecting which some doubt was perhaps entertained, by those who were not sufficiently acquainted with the few, but satisfactory experiments relating to it, which had before been made in this country; that is, upon the perfect accuracy of the Huygenian laws of refraction in the Iceland crystal, and other doubling substances; which, considered in itself, he thinks one of the finest discoveries of this celebrated geometrician.'

Newton,' he observes, was acquainted with the investigations of Huygens; yet he attempted to substitute, for the Huygenian law, another apparently more simple, but absolutely contrary to the phenomena, as Mr. Hauy first observed and demonstrated. It is difficult to explain the disregard that Newton showed, in this case, to a law which Huygens had declared to be conformable to his experiments.'

Wollaston has examined the refractive power of the crystal, by a very ingenious method of his own invention, and has shown that the law is perfectly true in all cases of rays passing in the direction of any surface of the crystal. He is the first, that after the oblivion of a century, thought of verifying, by direct experiments, a law which Huygens had considered as incontestible, and which Newton had rejected without exexamination.

It may not, however, be altogether superfluous to observe, that as Dr. Wollaston's experiments seem to have led to Mr. Malus's researches and discoveries, so Dr. Wollaston's thoughts were in all probability directed to the Huygenian theory by an earlier paper published in the same volume of the Philosophical Transactions with his own, in which it is asserted, almost in the terms that Mr. Malus has employed, that Newton, without attempting to deduce from his own system any explanation of the more universal and striking effects of doubling spars,-has omitted to observe, that Huygens's most elegant and ingenious theory perfectly accords with these general effects in all particulars.' Ph. Tr. 1802. 45. In short, whoever reads the account, which Huygens gives of his own examination of these substances, can scarcely fail to be convinced that his law must be extremely near the truth. Dr. Wollaston's experiments afforded additional evidence of its accuracy; and Mr. Malus, having diversified his calculations and observations in a still greater variety of forms, has left nothing further to be desired, for the complete re-establishment of this remarkable result of a hypothetical theory, the groundwork of which is still by no means unexceptionable notwithstanding the wonderful simplicity to which as we have shown on a former occasion (No. IV. p. 344), it is capable of being easily reduced.

Mr. Malus has prefixed to his experimental investigations an analytical treatise on optical phenomena in general, which will probably be thought, by most English readers, unnecessarily intricate, and which does not appear to contain any material novelty. He has examined the forms of the principal refracting substances by means of a reflective goniometer, resembling Dr. Wallaston's; and he has taken the mean of a considerable number of successive repetitions of the measurement. For the angle of the Iceland spar he obtains, in this manner, 74° 55′ 2′′; and contents himself with 74° 55', which is precisely Dr. Wollaston's measure: for the indices of refraction he gives 1.6543 and 1.4833 (p. 199), instead of Dr. Wollaston's 1.657 and 1.488, although the experiments on some specimens go as far as 1.658 (p. 105). For quartz crystal we have 1.5582 and 1.5484; for the sulfate of barita, 1.6468 and 1.6352; and for the arragonite, another form of the carbonate of lime, which some have suspected to contain strontia, 1.9631 and

1.5348. From the laws of extraordinary reflection within a crystal of doubling spar, Mr. Malus has very ingeniously deduced an explanation of a reduplication of images, long since observed by Martin, in particular specimens, which appear to have been interrupted by fissures, of such a nature, as to be capable of producing a subdivision of the rays, like that which takes place in the internal reflections: the colours observable in these images he refers to the thickness of the fissures, although it seems at least equally probable, that they are more nearly related to the colours of crystalized substances, since described by Biot and others.

Mr. Malus's calculations of the particular cases of refraction are founded on the Huygenian method of drawing a tangent plane to the supposed spheroid, from a point in the surrounding medium, at which the supposed original undulation would have arrived while the spheroid is generated. The steps of this mode of calculation are, however, extremely intricate; and it has occurred to us that the problem may be solved in a much more simple manner, by equating the velocities with which the supposed undulations must advance upon the common surface of the respective mediums: a condition which is obviously sufficient for the deter-t mination of the angular directions of the actual undulations; just as the velocity, with which a bird swims on the surface of a piece of water,is sufficient for determining the direction of the wave which follows it. Considering the velocity of the advance of the undulation with regard to the spheroid, it must evidently be identical with the velocity of increase or decrease of the perpendicular to the circumference of the section cut off by the refracting surface; and with regard to the surrounding space, it must be to the direct velocity, as the radius to the sine of the angle of incidence or refraction in that space. Hence if r be the index of the greatest refractive density of the substance, s the sine of incidence or refraction without the crystal, a the semiaxis of the spheroid, and y the perpendicular falling from the point of incidence on the conjugate diameter of the section, we have the equation rx: sy; which determines the physical conditions of the problem, and reduces it to a mathematical investigation.

Now if the ratio of the greatest and least refractive densities, or of the equatorial diameter of the spheroid to the axis, be that of n to 1, and the tangent of the angle formed by the axis with the refracting surface, p, it may readily be inferred, from comparing the ordinates of the ellipsis with those of the inscribed circle, and from the properties of similar triangles, that the semidiameter parallel 1+pp to the given surface will be n r, the tangent of the angle nn+pp formed by the conjugate semidiameter with the axis, nn: p,and the

-x. From the known equality

length of this semidiameter p2+na.

nn+pp

of all parallelograms described about an ellipsis, we have, for the perpendicular falling from the end of this semidiameter on the former,+PP; and, taking the difference of the squares,

p(un-1)x √(nn+pp)√(1+pp)'

for its distance from the centre; and for the

sine of the included angle,

dicular falling from the same point on the axis will be

(nn+pp)

Nnx

́√(nn+pp)=

=(v.)

Proceeding now to the section formed by the given refracting surface, let q be the cotangent of the angle comprehended by its lesser axis and the plane of the ray's motion without the crystal, and let z be the distance of its centre from that of the spheroid we shall then have, for the lesser semiaxis of the section, (p2+n4 x2-22), reduced in the ratio of the conjugate diameters

(p2 1

nn+pp

der to find y, we must substitute these values in the expression for the

perpendicular ✔ x, whence we have y

nn+pp
1+PP

=

‚mm+qq, [p2+n^ 1+qq nn+pp mm+qg

x2--z2 ) n1+; and taking the fluxion, rr: s=y=√

p2+ns 1+PP <p2 + n 4

terof the section ending at the point of incidence,

1+pp nn+pp

S

s = 92 + m2
╈*
n2 x
r 1+99

: whence it is obvious that this semidiameter, which may be

considered as an ordinate in an elliptic section passing through the centre of the spheroid, is proportional to the sine of incidence, as Huygens has demonstrated; and it will appear that the tangent of the angle formed by the plane of this section with the plane of incidence is P(n-1) mm

́nn +pp; √(q2+m*)*

But in order to determine more directly the inclination of the ray within the crystal, we must find u, the perpendicular falling from the point of incidence on the lesser semiaxis, before expressed mmnns 1+pp

nnx

by

√(nn+pp)

and now=

r√(1+qq) nn+pp'

the centre of the section v=

and its distance from

qnns +PP and from the r√(t+qq) nn+pp'

point nearest to the centre of the spheroid, tz-v; whence the distance of the point of incidence from this last point must be ✔([tz-v]+u): and adding to the square of this that of the perpendicular falling on the section from the centre of the spheroid,or 22-2, we have v (v tu+z2-2tvz) for the semidiameter at the point of incidence, expressing the velocity: and the sine of the corresponding angle of incidence or refraction will be ✓ ([tz-v]2 tu) divided by this radius, while that of the inclination of the plane to the axis will be u: √ ([tz—v]2+u2); z being=x√ rp2+n2 ss mm+qq. p2+n

4

nn+pp rr. 1+qq nn+pppp). It is also evident that the velocity, reduced to the direction of a perpendicular to the surface, will vary as √(1–12).

These expressions may be much simplified by further reduction, especially where they are to be applied to surfaces either parallel or perpendicular to the axis: since in these cases p0 and m=n, and poo and m 1 respectively, and t-0 in both. Hence, in

√(r2 — n2s2)°

n+qq ss

r(1+qq),2; r√ (1+qq)

whence the sine of the angle may

(u+v+2); and the tangent will be

nn+qq 1+99 n2 sx

It is not merely with a view of exhibiting a more convenient mode of solving a problem which Mr. Malus had solved before,